Technically open source and free software more or less mean the same thing. They have different definitions, but they result basically in the same thing. So why was the Open Source Initiative founded and why did they coin the term open source? What are the differences they intended to make? How do the philosophies behind the two camps differ?
One of the most helpful analogies I've ever heard compares software development to government:
Lots of people like democratic governance. Some people like democracy because it produces the best results that maximize happiness of its citizens. Other people like democracy because they believe people have a moral right to have their voices be part of their own government (regardless of the results). The first group is like open source; the second group is like free software.
Richard Stallman coined the term "Free Software" to mean freely-shareable and freely-modifiable software that complies with his four freedoms. The Free Software Foundation has always been an ethically-driven organization. In the documentary Revolution OS, Richard Stallman, founder of the FSF, says of the GNU project, (emphasis mine):
And to me, [a non-disclosure agreement] was essentially a promise to be a bad person... to cut myself off from... a cooperating community. [...] [By developing GNU,] not only could I give myself a way to keep using computers without betraying other people, but I'd give to everybody else, too. Everybody else would have a way out of that moral dilemma...
The term "Open Source Software" was invented later, in a meeting between Eric S. Raymond, some employees of VA Linux Systems, and a few others. There's an interview with Eric S. Raymond in Revolution OS in which he recalls the original effort to develop and popularize the term:
[Y]ou walk into an executive's office and say "Free Software"... Okay, if you're lucky, the response is something like, "Hmm, Free Software, must be cheap, shoddy, worthless." And if you're not lucky, it has associations with the Free Software Foundation's wholesale attack on intellectual property rights, which -- regardless of what you think about the ethics of that -- it's lousy marking. It's not something businesses want to hear.
There's a following clip from Larry Augustin, CEO of VA Linux Systems:
We'd been calling this "Free Software" but people took the term "Free" and associated with "free of charge". They thought they couldn't make money or couldn't sell. We wanted to get across the idea that the software was open and that the source code was available. Very important pieces.
It is clear from these interviews, "open source" was intended to capture the idea of freely-modifiable and shareable software in a business-centered way, rather than as an ethical concern. In addition, the confusion over "free in cost" was a point of concern for business adoption as well.
For Stallman (and by extension, the whole FSF, which he founded), the freedom to modify and share software has always been a moral issue. The Open Source movement began as an attempt to capitalize on the practical benefits of freely-shareable and modifiable software without the ethical concerns that have always been at the core of the Free Software movement.
In practical terms, "open source software" means "software complying with the OSI's Open Source Definition," while "free software" means "software complying with the FSF's four freedoms." The two definitions are not perfectly equivalent, but they are very close.
The Open Source Initiative can sort of be thought of the branch of Free Software that became its own tree.
The term "free software" is older, and is reflected in the name of the Free Software Foundation (FSF), an organization founded in 1985 to protect and promote free software. The term "open source" was coined in 1998 by a group of people — the founders of the Open Source Initiative (OSI) — who also supported the development and distribution of free software, but who disagreed with the FSF about how to promote it, and who felt that software freedom was primarily a practical matter rather than an ideological one... Source
So those involved with Open Source supported the development and distribution of the software. They branched out because they disagree with the idea that software freedom is ideological; they thought that software freedom was primarily a practical matter.
The page goes on to say that they had differences in why they should promote the software, and how to promote it, even though the ideas were incredibly similar.
[Not] surprisingly, one of the biggest concerns was the use of the word "Free." For this reason, we see a lot of French in our English. :) Je suis libre!
Bottom line, Open Source originated from people who supported the Free Software Movement, but disagreed with promotion and overall philosophy.